You get 1 free mulligan at the beginning of each match. Use this to your advantage if you have a sub-par beginning hand. Feel free to restart the match while in campaign if you get a poor starting hand as this will cut down on wasted time towards unlocking all the cards.

You unlock cards by beating ANY opponent, except for Challenges. So, you can unlock cards by repeatedly playing against Chandra Nalaar at the very beginning of the campaign, even though the AI will constantly adjust to your play styles and get increasingly difficult. If you choose to do this, the deck you will fight against is always the same so there's no need to be cautious of cards that would be unlocked in later duels.

Always try to act LAST in each altercation. The chance to go last means that your spell/ability will resolve before the spells/abilities that are already being played. This is known as FILO (First in, Last Out) and using this to your advantage will make many of the more difficult tasks easier.

If you're interested in unlocking the cards as quickly as possible, you can find a friend and take turns in Player Matches to unlock cards. You can 'Concede Game' to give your opponent a quick win.

Tsunami of Pain "Deal 12 combat damage to an opponent in a single attack with one blue creature."

There are several ways to accomplish this, but I'll list two relatively easy ways. (card images in spoiler)

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Using the deck Relics of Doom, you can exploit Master of Etherium's ability to allow him to grow to a gigantic monster. This deck is full of artifacts, so he'll be quite large by the time you reach Sharding Sphinx. The Sphinx's ability will allow you a near limitless amounts of artifacts, as long as you're able to deal damage to a player. Once your Master of Etherium is (minimum) attack power of 12, attack your opponent (hopefully, he can get through unblocked) and you'll unlock this trophy.

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You can also use either Thoughts of Wind or Root of the Firemind and use Sphinx of Magosi's ability until he's 12/12 or larger.

These strategies can be used in offline duels against the AI, or boosted online with a partner.

Salt in the Wound
"An opponent's last permanent leaves the battlefield."

This sounds difficult, but it's quite simple using the Thoughts of Wind deck, which has Boomerang.

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Make sure you have a Boomerang in your opening hand, and at least two Islands. You have to go first, so restart the duel if you don't have the proper hand, or if you're not going first. Simply play an Island, end your turn, and as long as your opponent doesn't play anything but a land card, you can play a second Island on your next turn, and use Boomerang to send their land back to their hand, earning you this trophy.